3D Vectors

All 3D vectors can be represented by a directed line segment in 3D space R3, which has a start point and an end point. This gives each vector a magnitude (the length of the line segment) and direction (from the start point to the end point). They have a rigorous definition in terms of vector spaces, but this'll do for now.
If we take the origin as out starting point we can describe any vector, a, by specifying its end point in cartesian coordinates (x, y, z). So any vector can be described by an ordered 3-tuple (x, y, z) where x, y and z are real numbers. In n dimensional space we could describe any vector by an ordered n-tuple (x1, x2, ... , xn) but we'll stay in 3D. Definition: The position vector of a point A is the vector represented by the line segment from the origin to A, and is written a.